On the compatible weaklynonlocal Poisson brackets of Hydrodynamic Type
Abstract
We consider the pairs of general weakly nonlocal Poisson brackets of Hydrodynamic Type (Ferapontov brackets) and the corresponding integrable hierarchies. We show that under the requirement of nondegeneracy of the corresponding "first" pseudoRiemannian metric $g_{(0)}^{\nu\mu}$ and also some nondegeneracy requirement for the nonlocal part it is possible to introduce a "canonical" set of "integrable hierarchies" based on the Casimirs, Momentum functional and some "Canonical Hamiltonian functions". We prove also that all the "Higher" "positive" Hamiltonian operators and the "negative" symplectic forms have the weakly nonlocal form in this case. The same result is also true for "negative" Hamiltonian operators and "positive" Symplectic structures in the case when both pseudoRiemannian metrics $g_{(0)}^{\nu\mu}$ and $g_{(1)}^{\nu\mu}$ are nondegenerate.
 Publication:

arXiv eprints
 Pub Date:
 November 2001
 DOI:
 10.48550/arXiv.nlin/0111015
 arXiv:
 arXiv:nlin/0111015
 Bibcode:
 2001nlin.....11015M
 Keywords:

 Exactly Solvable and Integrable Systems;
 Mathematical Physics
 EPrint:
 31 Pages, Latex, reference added